Embeddability of joins and products of polyhedra

TL;DR

This paper provides a new proof of a theorem on the embeddability of certain polyhedra and explores conditions under which joins and products of polyhedra embed in Euclidean spaces, highlighting the use of geometric cohomology.

Contribution

It offers a concise proof of Parsa's theorem and establishes new embeddability criteria for joins and products of polyhedra in Euclidean spaces.

Findings

A compact n-polyhedron P exists that does not embed in R^{2n} but P*P embeds in R^{4n+2}.

Embeddability of X in R^m is guaranteed under certain join and product conditions involving skeletons and acyclic complexes.

The proof showcases the application of geometric cohomology in embedding problems.

Abstract

We present a short proof of S. Parsa's theorem that there exists a compact $n$-polyhedron $P$, $n\ge 2$, non-embeddable in $\mathbb R^{2n}$, such that $P*P$ embeds in $\mathbb R^{4n+2}$. This proof can serve as a showcase for the use of geometric cohomology. We also show that a compact $n$-polyhedron $X$ embeds in $\mathbb R^m$, $m\ge\frac{3(n+1)}2$, if either - $X*K$ embeds in $\mathbb R^{m+2k}$, where $K$ is the $(k-1)$-skeleton of the $2k$-simplex; or - $X*L$ embeds in $\mathbb R^{m+2k}$, where $L$ is the join of $k$ copies of the $3$-point set; or - $X$ is acyclic and $X\times\text{(triod)}^k$ embeds in $\mathbb R^{m+2k}$.